But he did say that every other recommendation of the landmark statement was "on the table" to be considered by a joint select committee of parliament along with the recommendations of previous work that went back to 2012.

Indigenous Labor senator Pat Dodson was not impressed.

"It's a bit like shut the gate after the bull's gone through isn't it?" he told the minister.

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Labor leader Bill Shorten told the prime minister the opposition was deeply disappointed with cabinet's decision to reject the "clear and unambiguous" call for a First Nations People voice without any meaningful work.

"It strikes me that a unilateral decision of this kind runs contrary to your repeated promise to do things with indigenous Australians, and not to them," he wrote in a letter to Malcolm Turnbull on Friday.

"It is not for us to dictate to First Nations Australians what form their recognition should take."

Indigenous advocate Noel Peaerson. (Photo:AAP). ()

Cape York indigenous leader Noel Pearson was scathing of the way cabinet's decision was made public late on Thursday, after parliament rose for the week.

"There could have been a way to say no to this, without all the egregious dog whistling that is present in the prime minister's press release," he told ABC radio.

Greens senator Rachel Siewert said the decision made a "mockery" of the government's claim that they listen to indigenous people.

Josephine Crawshaw, who is involved in the Uluru statement group, said there was a sense of "groundhog day" for indigenous people.

Senator Scullion didn't disagree, telling the upper house committee: "It is sad, that we appear to be starting again."

The National Congress of Australia's First Peoples said the decision was sending "shockwaves" through indigenous communities.